Monitoring system for well casing

ABSTRACT

A system ( 20 ) for monitoring deformation of a substantially cylindrical casing ( 14 ). The system ( 20 ) includes at least two strings ( 22 ) of interconnected sensors ( 24 ) that are wrapped around the casing ( 14 ) so as not to intersect one another. At least one of the strings ( 22 ) includes a series of at least two segments (S). The series of segments (S) includes a segment (S) arranged at one wrap angle (θ) and another segment (S) arranged at a different wrap angle (θ).

TECHNICAL FIELD

This invention relates generally to systems and methods for detectingdeformation of a casing of a well in a formation and, more specifically,to a system that includes strings of interconnected strain sensors.

BACKGROUND

Electromagnetic investigation tools are often used to take measurementsat points along the length of a borehole in an earth formation. Wells informations are commonly reinforced with casings that prevent the wellsfrom collapsing. However, forces applied by the formation may cause thecasing to bend, buckle, or otherwise deform. Where the deformationresults in a significant misalignment of the well axis, the productionthat can be gained from the well can may be partially or completelylost. In either case, additional time and expense is necessary to repairor replace the well.

The ability to detect an early stage of deformation would allow forchanges in production practices and remedial action.

SUMMARY

The present disclosure provides a system and method for detectingdeformation of a casing in a formation. The system includes nonintersecting strings of interconnected sensors such that the risk ofdamage is reduced. The strings are arranged to facilitate qualitativeand/or quantitative analysis of data from the interconnected sensors.

According to an exemplary embodiment, a system for monitoringdeformation of a substantially cylindrical casing includes at least twostrings of interconnected sensors. The strings are wrapped around thecasing so as not to intersect one another. The strings include a firststring that includes a first series of at least two segments. The firstseries of at least two segments includes a first segment arranged at afirst angle with respect the casing axis and a second segment arrangedat a second angle with respect to the casing axis.

In certain embodiments, first series of at least two segments furtherincludes a third segment arranged at a third angle. In certainembodiments, the strings include a second string that is arranged at asubstantially constant third angle. In certain embodiments, the segmentsextend for arc distances that are at least half of the circumference ofthe casing.

Grooves are formed in the casing and the strings are at least partiallyrecessed in the grooves.

The system further includes a data acquisition unit and a computing unitfor collecting and processing data measured at the sensors. In certainembodiments, at least one of the interconnected sensors measures strain.In certain embodiments, at least one of the interconnected sensorsmeasures temperature.

According to one aspect of the disclosure, the strings include a secondstring that includes a second series of at least two segments. Thesecond series of at least two segments includes a third segment that isarranged at a third angle with respect the casing axis and a fourthsegment arranged at a fourth angle with respect to the casing axis.

In certain embodiments, the first series of at least two segments issubstantially the same as the second series of at least two segments.According to an exemplary embodiment, axial distances of the segmentsare substantially equal to one another. In such embodiments, the firststring and the second string can be positioned relative to one anothersuch that segments that have different wrap angles are representedwithin distance intervals along the axial length of the casing.According to another exemplary embodiment, arc distances of the segmentsare substantially equal to one another. In such embodiments, the firststring and the second string can be positioned relative to one anothersuch that segments that have the same wrap angle are represented withindistance intervals along the axial length of the casing.

According to another aspect of the disclosure, the strings includeoptical fibers and the sensors include periodically written wavelengthreflectors. In certain embodiments, the wavelength reflectors arereflective gratings such as fiber Bragg gratings.

In such embodiments, strings provide a wavelength response that includesreflected wavelengths corresponding to sensors. Each reflectedwavelength is substantially equal to the sum of a Bragg wavelength and achange in wavelength. The change in wavelength corresponds to a strainmeasurement.

Strings can be arranged such that subsets of the wavelength responsescan be grouped according to wrap angle and such that at least one of thegrouped subsets includes substantially continuous measurements along thelongitudinal axis of the casing. Strings can also be arranged such thatsubsets of the wavelength responses can be grouped according to wrapangle and such that at least one of the grouped subsets includessubstantially continuous measurements along the circumference of thecasing.

According to another aspect of the disclosure, a method of imagingdeformation of a cylindrical casing includes measuring an amount ofstrain at a plurality of positions on a casing, determining thedeformation of the casing from the strain measurements, and projectingan image of the deformed casing. The strain is measured by receivingsignals from at least two strings of interconnected sensors that arewrapped around the casing so as not to intersect one another. At leastone of the strings includes a series of at least two segments. Theseries of at least two segments includes a first segment arranged at afirst angle with respect the casing axis and a second angle arranged ata second angle with respect to the casing axis. A memory or computerreadable medium includes computer executable instructions for executionof the method.

According to another aspect of the invention, a cylindrical casingincludes at least two grooves for receiving at least two strings ofinterconnected sensors. The grooves are arranged so as not to intersectone another. At least one of the grooves includes a series of at leasttwo segments. The series of at least two segments includes a firstsegment arranged at a first angle with respect the casing axis and asecond segment arranged at a second angle with respect to the casingaxis.

The foregoing has broadly outlined some of the aspects and features ofthe present invention, which should be construed to be merelyillustrative of various potential applications of the invention. Otherbeneficial results can be obtained by applying the disclosed informationin a different manner or by combining various aspects of the disclosedembodiments. Accordingly, other aspects and a more comprehensiveunderstanding of the invention may be obtained by referring to thedetailed description of the exemplary embodiments taken in conjunctionwith the accompanying drawings, in addition to the scope of theinvention defined by the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a partial cross-sectional view of a well reinforced with acasing.

FIG. 2 is a partial side view of the casing of FIG. 1 and a system formeasuring deformation of the casing.

FIGS. 3-7 illustrate exemplary arrangements of strings of the system ofFIG. 2.

FIG. 8 is a graph illustrating signals relating to the arrangement ofstrings shown in FIG. 3.

FIG. 9 is a graph illustrating signals relating to the arrangement ofstrings shown in FIG. 4.

FIG. 10 is a graph illustrating signals relating to the arrangement ofstrings shown in FIG. 5.

FIG. 11 is a graph illustrating signals relating to the arrangement ofstrings shown in FIG. 6.

DETAILED DESCRIPTION OF THE INVENTION

As required, detailed embodiments of the present invention are disclosedherein. It must be understood that the disclosed embodiments are merelyexemplary of the invention that may be embodied in various andalternative forms, and combinations thereof. As used herein, the word“exemplary” is used expansively to refer to embodiments that serve asillustrations, specimens, models, or patterns. The figures are notnecessarily to scale and some features may be exaggerated or minimizedto show details of particular components. In other instances, well-knowncomponents, systems, materials, or methods have not been described indetail in order to avoid obscuring the present invention. Therefore,specific structural and functional details disclosed herein are not tobe interpreted as limiting, but merely as a basis for the claims and asa representative basis for teaching one skilled in the art to variouslyemploy the present invention.

Systems and methods are described herein in the context of determiningdeformation of a well casing. However, the present disclosure is alsoapplicable to other cylindrical objects in a borehole where the systemsand methods are used to detect and monitor deformation of the objectduring production or other non-production operations such as completion,gravel packing, frac packing, production, stimulation, and the like. Thecylindrical objects may be in the form of a well bore tubular, a drillpipe, a production tube, a casing tube, a tubular screen, a sand screen,and the like.

The teachings of the present disclosure may also be used in otherenvironments where pipes expand, contract, or bend. Examples of suchenvironments include refineries, gas plants, and pipelines.

Herein, a suffix (a, b, c, etc.) or subscript (1, 2, 3, etc.) is affixedto an element numeral that references like-elements in a general mannerso as to differentiate a specific one of the like-elements. For example,groove 30 a is a specific one of grooves 30.

Casing

Referring to FIG. 1, a well 10 is drilled in a formation 12. To preventwell 10 from collapsing or to otherwise line or reinforce the well 10, acasing 14 is formed in well 10. In the exemplary embodiment, casing 14is formed from steel tubes that are inserted into well 10. Cement ispoured between casing 14 and formation 12 to provide a bonded cementsheath 16. However, in alternative embodiments, casing 14 may be formedfrom other materials and according to alternative methods.

For purposes of teaching, coordinate systems are now described. ACartesian coordinate system can be used that includes an x axis, a yaxis, and a z axis that are orthogonal to one another. The z axiscorresponds to the longitudinal axis of casing 14 and any position oncasing 14 can be established according to an axial position z and aposition in the x-y plane, which is perpendicular to the z axis. In theillustrated embodiment, casing 14 is cylindrical and any position oncasing 14 can be established using a Cylindrical coordinate system.Here, the z axis is the same as that of the Cartesian coordinate systemand a position lying in the x-y plane is represented by a radius r and aposition angle α and referred to as a radial position rα. Radius rdefines a distance of the radial position rα from the z axis and extendsin a direction determined by position angle α to the radial position rα.Here, position angle α is measured from the x axis.

A bending direction represents the direction of a bending moment oncasing 14. The bending direction is represented by a bending angle βthat is measured relative to the x axis. A reference angle φ is measuredbetween bending angle β and position angle α.

Deformation

Casing 14 may be subject to forces, such as shear forces and compactionforces exerted, for example, by formation 12 or by the inflow of fluidbetween formation 12 and casing 14. These forces can cause casing 14 todeform. An example of a force F causing deformation of casing 14 isillustrated in FIG. 2.

System

Continuing with FIG. 2, casing 14 includes a system 20 for detectingdeformation. System 20 includes strings 22 of interconnected strainsensors 24 that are wrapped around casing 14 such that sensors 24 arepositioned along the axial length and circumference of casing 14.

System 20 further includes a data acquisition unit 38 and a computingunit 40. Data acquisition unit 38 collects the response at the sensors24 of each of the strings 22. The response and/or data representativethereof are provided to computing unit 40 to be processed. Computingunit 40 includes computer components including a data acquisition unitinterface 42, an operator interface 44, a processor unit 46, a memory 48for storing information, and a bus 50 that couples various systemcomponents including memory 48 to processor unit 46.

Strings of Interconnected Sensors

There are many different suitable types of strings 22 of interconnectedsensors 24 that can be associated with system 20. For example, strings22 can be waveguides such as optical fibers and sensors 24 can bewavelength-specific reflectors such as periodically written fiber Bragggratings (FBG). An advantage of optical fibers with periodically writtenfiber Bragg gratings is that fiber Bragg gratings are less sensitive tovibration or heat and consequently are far more reliable. In alternativeembodiments, strain sensors 24 can be other types of gratings,semiconductor strain gages, piezoresistors, foil gages, mechanicalstrain gages, combinations thereof, and the like.

Sensors 24 are not limited to strain sensors. Rather, in certainapplications, sensors 24 are temperature sensors.

According to a first exemplary embodiment described herein, strings 22are optical fibers and sensors 24 are fiber Bragg gratings.

A wavelength response λ_(n) of a string 22 is data representingreflected wavelengths λ_(r) at sensors 24. The reflected wavelengthsλ_(r) each represent a fiber strain ε_(f) measurement at a sensor 24.

Generally described, reflected wavelength λ_(r) is substantially equalto a Bragg wavelength λ_(b) plus a change in wavelength Δλ.Specifically, reflected wavelength λ_(r) is equal to Bragg wavelengthλ_(b) when fiber strain ε_(f) measurement is substantially zero. Whenfiber strain ε_(f) measurement is non-zero, reflected wavelength λ_(r)differs from Bragg wavelength λ_(b). The difference is change inwavelength Δλ and thus change in wavelength Δλ is the part of reflectedwavelength λ_(r) that is associated with fiber strain ε_(f). Braggwavelength λ_(b) provides a reference from which change in wavelength Δλis measured at each of sensors 24. The relationship between change inwavelength Δλ and fiber strain ε_(f) is described in further detailbelow.

Multiple Strings and Multiple Wrap Angles

In the illustrated embodiments, system 20 includes a plurality ofstrings 22 and each string 22 winds substantially helically at leastpartially along the length of casing 14. Certain of strings 20 include aseries of segments S that are arranged at different inclinations,hereinafter referred to as wrap angles θ. Typically, the series is atleast partially repeated.

In general, wrapping strings 22 at an angle is beneficial in thatstrings 22 only experience a fraction of the strain experienced bycasing 14. Each wrap angle θ is effective for a range of strain.Accordingly, the use of multiple strings 22 with different wrap angles θexpands the overall range of strain that system 20 can measure. Forexample, a string with a wrap angle of 20° may fail at one level ofstrain while the same string with a wrap angle of 30° or more may notfail at the same level of strain or at a slightly higher level ofstrain.

The use of multiple strings 22 with different wrap angles θ facilitatesdetermining Poisson's ratio v, as described in further detail below.Poisson's ratio v may be an undetermined parameter where casing 14nonelastically deforms or yields under higher strains. For example,where casing 14 is steel, Poisson's ratio v may be near 0.3 whiledeformation is elastic, but trends toward 0.5 after deformation becomesnon-elastic and the material yields.

Another advantage of wrapping casing 14 with multiple strings 22 is thatthere is added redundancy in case of failure of one of strings 22. Theadditional data collected with multiple strings 22 makes recovery of a3-D image an overdetermined problem thereby improving the quality of theimage.

Non Intersecting Method of Wrapping Multiple Strings at Multiple Angles

In the illustrated embodiments, strings 22 are arranged so as not tointersect one another. Referring to FIG. 2, grooves 30 are formed incasing 14 and strings 22 are at least partially recessed in grooves 30.As strings 22 are arranged so as to not intersect one another, the depthof grooves 30 is minimized and, accordingly, the effect of grooves 30 onthe integrity of casing 14 is minimized. Conversely, were strings 22 tobe arranged to intersect, at least part of the depth of grooves 30 wouldhave to be increased at regions of intersection so that strings 22 wouldnot protrude out of grooves 30. However, the increased depth of grooves30 would have a greater effect on the integrity of casing 14.Alternatively, if the depth of grooves 30 is not increased, overlappingstrings 22 would protrude outside grooves 30 thereby increasing the riskof being damaged.

Exemplary arrangements of strings 22 are now described. In general, thedescription of an arrangement of strings 22 is applicable to anarrangement of grooves 30 as strings 22 are received in grooves 30. Inother words, a string 22 and a corresponding groove 30 followsubstantially the same path.

Referring to FIGS. 3-7, casings 14 are shown in an unrolled or flattenedcondition to illustrate arrangements of strings 22. In other words,axial position z is plotted on the vertical axis and radial position rαis plotted on the horizontal axis.

Generally described, each casing 14 is wrapped with strings 22 that windsubstantially helically at least partially along the axial length ofcasing 14. At least one of strings 22 includes a series of segments Sthat are arranged at different inclinations or wrap angles θ. Theillustrated wrap angles θ are measured with respect to x-y planes thatare represented by notional dotted lines although equivalent alternativeformulations can be achieved by changing the reference plane used tomeasure wrap angles θ.

Segments S are arranged at wrap angles θ such that, as segments S arewrapped around casing 14, segments S longitudinally ascend an axialdistance L along the axial length of casing 14 and transversely extendan arc distance C around the circumference of casing 14.

As mentioned above, wrap angle θ can be selected according to a range ofstrains that system 20 is likely to encounter or designed to measure.The lengths of segments S may then be selected in any manner so long asstrings 22 do not intersect and overlap one another. Exemplary methodsfor selecting the lengths of segments S are now described. As describedin further detail below, the selection of the lengths of segments Sfacilitates qualitative and quantitative analysis of wavelengthresponses λ_(n).

Arrangements of Strings

Referring to FIGS. 3 and 4, the illustrated arrangements include firststring 22 a and second string 22 b where first string 22 a has arepeating series of segments S₁, S₂ and string 22 b has a repeatingseries of segments S₃, S₄. Strings 22 a, 22 b are substantially similarto one another as segments S₁, S₃ have substantially the same length andwrap angle θ and segments S₂, S₄ have substantially the same length andwrap angle θ. Specifically, wrap angle θ₁ is substantially equal to wrapangle θ₃ and wrap angle θ₂ is substantially equal to wrap angle θ₄.

The difference in position of strings 22 a, 22 b relative to oneanother, generally referred to herein as phase, is selected such thatthe strings do not intersect. Phase can be indicated by the distance anddirection between reference points p on strings 22. Reference points pmay be selected where a series of segments S begins or ends or at ameeting point of segments S. For example, referring to FIG. 3, areference point p on first string 22 a and a corresponding referencepoint p on second string 22 b have different axial positions z andradial positions rα. Referring to FIG. 4, a reference point p on firststring 22 a and a corresponding reference point p on second string 22 bhave the same axial position z and different radial positions rα.

For clarity, in FIGS. 3 and 4, first string 22 a is illustrated as arelatively thicker line, second string 22 b is illustrated as arelatively thinner line, segments S₁, S₃ are illustrated as solid lines,and segments S₂, S₄ are illustrated as dashed lines.

Referring to FIG. 3, lengths of segments S₁, S₂, S₃, S₄ are selectedsuch that axial distances L₁, L₂, L₃, L₄ are substantially the same andequal to constant distance intervals N measured along the axial lengthof casing 14. Phase is selected such that segments S₁, S₄ arerepresented within every other distance interval N and segments S₂, S₃are represented within other distance intervals N.

As described in further detail below, when wavelength responses λ_(n) ofboth strings 22 a, 22 b are plotted on the same graph with respect toaxial position z, subsets u of wavelength responses λ_(n) can be groupedaccording to wrap angle θ such that a group of subsets u represents asubstantially continuous series of measurements along the axial lengthof casing 14 for one wrap angle θ value. Referring momentarily to FIG.8, for the arrangement of FIG. 3, subsets u of wavelength responsesλ_(n1), λ_(n2) that correspond to wrap angles θ₁, θ₃ can be combined andsubsets of wavelength responses λ_(n1), λ_(n2) that correspond to wrapangles θ₂, θ₄ can be combined.

Referring to FIG. 4, lengths of segments S₁, S₂, S₃, S₄ are selectedsuch that arc distances C₁, C₂, C₃, C₄ are substantially the same.Specifically, each arc distance C₁, C₂, C₃, C₄ is substantially half ofthe circumference of casing 14.

Phase is selected such that segments S₁, S₃ are represented within everyother distance interval N and segments S₂, S₄ are represented withinother intervals N. Here, distance intervals N change in length in analternating manner according to different axial distances L₁, L₂, L₃,L₄.

Referring momentarily to FIG. 9, for the arrangement of FIG. 4, subsetsu of wavelength responses λ_(n1), λ_(n2) that correspond to wrap anglesθ₁, θ₃ can be combined and subsets u of wavelength responses λ_(n1),λ_(n2) that correspond to wrap angles θ₂, θ₄ can be combined. Here, thegroups of subsets u are interrupted and only partially represented alongthe axial length of casing 14 but effectively measure around the entirecircumference of casing 14.

In alternative embodiments, the lengths of segments S₁, S₂, S₃, S₄ canbe constrained so as to be substantially equivalent.

The teachings of the present disclosure are not limited to a systemhaving two strings 22 where each string 22 is arranged to include twowrap angles θ. Referring to FIGS. 5-7, embodiments of system 20 aredescribed that include at least two strings 22 where at least one ofstrings 22 is arranged to include inclinations of at least two wrapangles θ.

Referring to FIG. 5, system 20 includes first string 22 a and secondstring 22 b. Here, string 22 a has a repeating series of segments S₁, S₂with different wrap angles θ₁, θ₂. String 22 b has a substantiallyconstant wrap angle θ₃ although, for purposes of teaching, string 22 bis described as a series of segments S₃ that have the same wrap angleθ₃.

Segments S₁, S₂ extend axial distances L₁, L₂ and arc distances C₁, C₂that are determined by the lengths of segments S₁, S₂ wrap angles θ₁,θ₂. String 22 b has wrap angle θ₃ where notional segments S₃ extend anaxial distance L₃ that is substantially equal to the sum of axialdistances L₁, L₂ and extend an arc distance C₃ that is substantiallyequal to the sum of arc distances C₁, C₂. String 22 a effectively variesabout a constant angle of inclination along the length thereof and theconstant angle of inclination is substantially equal to wrap angle θ₃.Strings 22 a, 22 b are therefore approximately parallel to one anotheralthough phased such that variations of string 22 a from a parallel pathdo not cause strings 22 a, 22 b to intersect one another.

Referring to FIG. 6, system 20 includes three strings 22 a, 22 b, 22 cand each string 22 a, 22 b, 22 c includes the same series of segmentsS₁, S₂, S₃ although, for simplicity, only string 22 a is labeled.Similar to the arrangement of FIG. 3, lengths of segments S₁, S₂, S₃such that axial lengths L₁, L₂, L₃ are substantially the same. Strings22 a, 22 b, 22 c are phased such that reference points p on strings 22a, 22 b, 22 c have different axial positions z and the same radialposition rα.

The previously described arrangements of FIGS. 3-5 can includeadditional strings 22 arranged at one or more wrap angles. For example,referring to FIG. 7, string 22 c is added to the arrangement of FIG. 3.Here, string 22 c has a substantially constant wrap angle θ₅ that can bedetermined as described for wrap angle θ₃ for the arrangement of FIG. 5.

Relationship Between Change in Wavelength and Strain

An equation that may be used to relate change in wavelength Δλ and fiberstrain εf imposed on sensors 24 is given by Δλ=λ_(b)(1−Pe)Kε_(ζ). As anexample, Bragg wavelength λ_(b) may be approximately 1560 nanometers.The term (1 −P_(e)) is a fiber response which, fox example, may be 0.8.Bonding coefficient K represents the bond of sensor 24 to casing 14 and,for example, may be 0.9 or greater.

The fiber strain ε_(f) may be associated with strain at a sensor 24position on casing 14 according to

$ɛ_{f} = {{- 1} + \sqrt{{\sin^{2}{\theta \cdot \left( {1 - \left( {ɛ_{a} - \frac{r\; \cos \; \varphi}{R}} \right)} \right)^{2}}} + {\cos^{2}{\theta \cdot \left( {1 + {\nu \left( {ɛ_{a} - \frac{r\; \cos \; \varphi}{R}} \right)}} \right)^{2}}}}}$

Fiber strain ε_(f) measured by sensor 24 at a position on casing 14 is afunction of axial strain ε_(a) at the position, radius of curvature R atthe position, Poisson's ratio v, wrap angle θ of segment S on whichsensor 24 is located, and radial position which is represented in theequation by radius r and reference angle φ. Fiber strain ε_(f) ismeasured, wrap angle θ is known, and radius r is known. Poisson's ratiov is typically known for elastic deformation of casing 14 and unknownfor non-elastic deformation of casing 14. Radius of curvature R,reference angle φ, and axial strain ε_(a) are typically unknown and aredetermined through analysis of wavelength response λ_(n). Similarly,Poisson's ratio v can be determined through analysis of wavelengthresponse λ_(n) where Poisson's ratio v is unknown.

Analysis of Wavelength Response

Referring to FIGS. 8-11, wavelength responses λ_(n) of strings 22 areplotted on the same graph. These measurements represent fiber strainε_(f) measurements made at each sensor 24 by system 20. Here, wavelengthresponses λ_(n) are plotted with respect to axial positions z of sensors24 or along the longitudinal axis of casing 14.

As mentioned above, each reflected wavelength λ_(r) of wavelengthresponse λ_(n) is substantially equal to Bragg wavelength λ_(b) pluschange in wavelength Δλ. As change in wavelengthΔλ is dependent on wrapangle θ, a shift in wavelength response λ_(n) (tracked by dotted lines)is observed where sensors 24 in series are on segments S that arearranged at different wrap angles θ. For example, a shift is observedapproximately at axial positions z where segments S interface. Aspreviously mentioned, subsets u of wavelength responsesλ_(n) thatcorrespond to one wrap angle θ can be grouped together to effectivelyprovide information that would be provided by a string 22 wrapped at asingle wrap angle θ.

Generally described, in response to axial strain ε_(a) on casing 14,wavelength response λ_(n) is typically observed as a constant (DC) shiftfrom Bragg wavelength λ_(b). In response to bending of casing 14 thatcorresponds to a radius of curvature R, wavelength response λ_(n) istypically observed as a sinusoid (AC). A change in Poisson's ratio vmodifies both the amplitude of the axial strain ε_(a) shift and theamplitude of the sinusoids. In any case, signal processing can be usedto determine axial strain ε_(a), radius of curvature R, reference angleφ, and Poisson's ratio v at sensor 24 positions. Examples of applicablesignal processing techniques include inversion where a misfit isminimized and turbo boosting. The signal processing method can includeformulating wavelength response λ_(n) for one wrap angle as thesuperposition of a constant shift and a sinusoid.

FIG. 8 represents exemplary wavelength responses λ_(n1), λ_(n2) measuredby system 20 where strings 22 a, 22 b are arranged as shown in FIG. 3.Here, wavelength responses λ₁, λ₂ are unique for axial strain ε_(a),radius of curvature R, and Poisson's ratio v.

FIG. 9 represents exemplary wavelength responses λ_(n1), λ_(n2) measuredby system 20 where strings 22 a, 22 b are arranged as shown in FIG. 4.Here, wavelength responses λ_(n1), λ_(n2) are unique for radius ofcurvature R. Specifically, subsets u within one of distance intervals Nspread apart with decreasing radius of curvature R.

FIG. 10 represents exemplary wavelength responses λ_(n1), λ_(n2)measured by system 20 where strings 22 a, 22 b are arranged as shown inFIG. 5. As wrap angle θ of string 22 b is substantially constant, thereis no shift due to change in wrap angle θ.

FIG. 11 represents exemplary wavelength responses λ_(n1), λ_(n2), λ_(n3)measured by system 20 where strings 22 a, 22 b, 22 c are arranged asshown in FIG. 6. The result is similar to that of FIG. 8 however thisarrangement provides three groups of subsets u corresponding to threedifferent wrap angle θ values.

The law does not require and it is economically prohibitive toillustrate and teach every possible embodiment of the presentdisclosure. Hence, the above-described embodiments are merely exemplaryillustrations of implementations set forth for a clear understanding ofthe principles of the invention. Variations, modifications, andcombinations may be made to the above-described embodiments withoutdeparting from the scope of the claims. All such variations,modifications, and combinations are included herein by the scope of thisdisclosure and the following claims.

What is claimed is:
 1. A system (20) for monitoring deformation of asubstantially cylindrical casing (14), comprising: at least two strings(22) of interconnected sensors (24) being wrapped around the casing (14)so as not to intersect one another, the at least two strings (22)comprising: a first string (22) comprising a first series of at leasttwo segments (S), the first series of at least two segments comprising:a first segment (S) arranged at a first angle (θ) with respect thecasing axis; and a second segment (S) arranged at a second angle (θ)with respect to the casing axis.
 2. The system (20) of claim 1, whereinthe first series of at least two segments (S) further comprises a thirdsegment (S) arranged at a third angle (θ).
 3. The system (20) of claim1, wherein the at least two strings (22) comprises a second string (22)arranged at a substantially constant third angle (θ).
 4. The system (20)of claim 1, wherein arc distances (C) of the segments (S) are at leasthalf of the circumference of the casing (14).
 5. The system (20) ofclaim 1, wherein grooves (30) are formed in the casing (14) and the atleast two strings (22) are at least partially recessed in the grooves(30).
 6. The system (20) of claim 1, wherein at least one of theinterconnected sensors (24) measures strain.
 7. The system (20) of claim1, wherein at least one of the interconnected sensors (24) measurestemperature.
 8. The system (20) of claim 1, wherein the at least twostrings (22) comprises: a second string (22) comprising a second seriesof at least two segments (S), the second series of at least two segments(S) comprising: a third segment (S) arranged at a third angle (θ) withrespect the casing axis; and a fourth segment (S) arranged at a fourthangle (θ) with respect to the casing axis.
 9. The system (20) of claim8, wherein the first series of at least two segments (S) issubstantially the same as the second series of at least two segments(S).
 10. The system (20) of claim 9, wherein axial distances (L)corresponding to the segments (S) are substantially equal to oneanother.
 11. The system (20) of claim 10, wherein the first string (22)and the second string (22) are positioned relative to one another suchthat segments (S) having different wrap angles (θ) are representedwithin distance intervals (N) along the axial length of the casing (14).12. The system (20) of claim 11, wherein arc distances (C) correspondingto the segments (S) are substantially equal to one another.
 13. Thesystem (20) of claim 12, wherein the first string (22) and the secondstring (22) are positioned relative to one another such that segments(S) having the same wrap angle (θ) are represented within distanceintervals (N) along the axial length of the casing (14).
 14. The system(20) of claim 1, wherein the at least two strings (22) include opticalfibers and the sensors (24) include periodically written wavelengthreflectors.
 15. The system (20) of claim 14, wherein each of the atleast two strings (22) provide a wavelength response (λ_(n)) thatcomprises reflected wavelengths (λ_(r)) corresponding to sensors (24),each reflected wavelength (λ_(r)) being substantially equal to the sumof a Bragg wavelength (λ_(b)) and a change in wavelength (Δλ), thechange in wavelength (Δλ) corresponding to a strain measurement (ε_(f)).16. The system (20) of claim 15, wherein the at least two strings (22)comprises a second string (22) comprising a second series of at leasttwo segments (S), the second series of at least two segments (S) beingsubstantially the same as the first series of at least two segments (S).17. The system (20) of claim 16, wherein the at least two strings (22)are arranged such that subsets of the wavelength responses (λ_(n)) canbe grouped according to wrap angle (θ) and such that at least one of thegrouped subsets (u) includes substantially continuous measurements alongthe longitudinal axis of the casing (14).
 18. The system (20) of claim17, wherein the at least two strings (22) are arranged such that subsetsof the wavelength responses (λ_(n)) can be grouped according to wrapangle (θ) and such that at least one of the grouped subsets (u) includessubstantially continuous measurements along the circumference of thecasing (14).
 19. A method of imaging deformation of a cylindrical casing(14), comprising: measuring an amount of strain at a plurality ofpositions on a casing (14), by: receiving signals from at least twostrings (22) of interconnected sensors (24) that are wrapped around thecasing (14) so as not to intersect one another, at least one of the atleast two strings (22) comprising a series of at least two segments (S),the series of at least two segments comprising: a first segment (S)arranged at a first angle (θ) with respect the casing axis; and a secondsegment (S) arranged at a second angle (θ) with respect to the casingaxis; determining the deformation of the casing (14) from the strainmeasurements; and projecting an image of the deformed casing (14).
 20. Acylindrical casing (14), comprising: at least two grooves (30) forreceiving at least two strings (22) of interconnected sensors (24), theat least two grooves (30) being arranged so as not to intersect oneanother; at least one of the at least two grooves (30) comprising aseries of at least two segments (S), the series of at least two segments(S) comprising: a first segment (S) arranged at a first angle (θ) withrespect the casing axis; and a second segment (S) arranged at a secondangle (θ) with respect to the casing axis.